G30 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



— crn . (Oo)r HOW has two branches for cr > —ao, which pass through the 



point a = — Co , X2 = — — making there a finite angle with each other. 



(Oo)r is completed by a loop in a < — o-q , Fig. 9(g), which does not 

 affect the incipient modes appreciably. The mode in question now has 

 two branches. The first starts as before and ends at or == — cro , where the 

 associated /3" is given by (3a- = K/o'o and \a is the smaller root of 



It resumes at <r = — o-q and (3" = fib = Xb/co , where Xb is the larger 

 root of the above equation, progresses to smaller | a |-values and then 

 back to 0- = — ao where fi' tends to infinity again in accordance Avith 

 (52). Be3'0nd/'o = Wi , where Oo rises steadily from o- = — x too- == with 

 increasing X, one branch of (Oo)r in — ao < c < disappears and only 

 the second branch of the mode remains. Neither branch has an analogue 

 in ordinary waveguides; as p -^ each lies in a smaller and smaller 

 neighborhood of o- = 1 , and finally vanishes into o- = 1 , X2 = — 1 . 



For n between Ui and ji there is a single solution curve starting at 

 (7 = and ending with /3" = at {I a)t — Ib ■ This may be identified 

 in the limit p = 0, with the TEn-limit mode, and has already been fully 

 discussed for H\ < rn < ji . No change in the formula for its cut-off point 

 occurs up to To = t<2 . A useful spot-point (Ib' — (Ii)t) along its course 

 can be found when 



< p < 1 - 4/ 1 _ -Zi: 



and is given by (60). 



In the range ji < ro < W2 , a further solution curve (corresponding to 

 the T]\Iii-limit mode) can arise, provided 



V < 4/1 -^ 



The radius at which it will then first appear is 



^0 



Vi - p- 



It begins on o- = 0, according to (39) and is cut off, with /3" = 0, at 

 As ro passes ih , the cut-off point of the TEi solution curve moves dis- 



